Topological Strings, Flat Coordinates and Gravitational Descendants

TL;DR

This paper explores the BRST formalism of topological minimal models coupled to gravity, constructs gravitational descendants, and derives recursion relations for genus-zero correlation functions, linking topological strings to integrable systems.

Contribution

It introduces a homotopy transformation simplifying the BRST operator, enabling explicit construction of gravitational descendants and connecting them to higher-order KdV flows.

Findings

Explicit construction of gravitational descendants

Derivation of recursion relations in topological gravity

Connection between topological strings and KdV integrable hierarchy

Abstract

We discuss physical spectra and correlation functions of topological minimal models coupled to topological gravity. We first study the BRST formalism of these theories and show that their BRST operator $Q=Q_s+Q_v$ can be brought to $Q_s$ by a certain homotopy operator $U$, $UQU^{-1}=Q_s$ ($Q_s$ and $Q_v$ are the $N=2$ and diffeomorphism BRST operators, respectively). The reparametrization (anti)-ghost $b$ mixes with the supercharge operator $G$ under this transformation. Existence of this transformation enables us to use matter fields to represent cohomology classes of the operator $Q$. We explicitly construct gravitational descendants and show that they generate the higher-order KdV flows. We also evaluate genus-zero correlation functions and rederive basic recursion relations of two-dimensional topological gravity.